Expectations from the Number and Operations Standard

Expectations from the Number and Operations Standard

Grades Pre-K-5

Compute fluently and make reasonable

estimates.

Grades Pre-K-2

• Develop and use strategies for whole-number

computations, with a focus on addition and

subtraction.

• Use a variety of methods and tools to compute,

including objects, mental computation,

estimation, paper and pencil, and calculators.

Grades 3-5

• Develop and use strategies to estimate the

results of whole-number computations and to

judge the reasonableness of such results.

• Select appropriate methods and tools for

computing with whole numbers from among

mental computation, estimation, calculators,

and paper and pencil according to the context

and nature of the computation and use the

selected method or tool.

Principles and Standards for School Mathematics

National Council of Teachers of Mathematics

2000

Master 10-1: Number and Operations Excerpts (Computational Methods)

Expectations from the Number and Operations Standard

Grades Pre-K-5

Understand numbers, ways of

representing numbers, relationships

among numbers, and number systems.

Grades Pre-K-2

• Develop a sense of whole numbers and

represent and use them in flexible

ways, including relating, composing,

and decomposing numbers.

Grades 3-5

• Develop and use strategies to estimate

the results of whole-number

computations and to judge the

reasonableness of such results.

• Develop fluency in adding, subtracting,

multiplying, and dividing whole

numbers.

Principles and Standards for School Mathematics

National Council of Teachers of Mathematics

2000

Master 10-2: Number and Operations Excerpts (Operation Sense)

A calculator should be used as a computational tool when it:

¤ facilitates problem solving

¤ eases the burden of doing tedious

computation

¤ focuses attention on meaning

¤ removes anxiety about doing

computation incorrectly

¤ provides motivation and confidence

A calculator should be used as an instructional tool when it:

¤ facilitates a search for patterns

¤ supports concept development

¤ promotes number sense

¤ encourages creativity and exploration

Master 10-3: The Calculator as a Tool

Suppose that you are an elementary school teacher that is involved in constructing

questions for a test. You want each question used to measure the mathematical

understanding of your students. For each proposed test item below, decide if

students should (S) use a calculator, it doesn't matter (DM) if the students use a

calculator, or students should not (SN) use a calculator in answering the test item

presented.

Does Should

ShouldNot Matter Not

A. 36 x 106 = . S DMSN

B. Explain a rule that generates S DMSN

this set of numbers:

..., 0.0625, 0.25, 1, 4, 16, ...

C. 12 - (8 - 2 x (4 + 3)) = . S DMSN

D. The decimal fraction 0.222 S DMSN

most nearly equals:

(a)

E. The number of students in S DMSN

each of five classes is 25,

21, 27, 29, and 28. What is

the average number of

students in each class?

F. I have four coins; each S DMSN

coin is either a penny,

a nickel, a dime, or a quarter.

If altogether the coins are

worth a total of forty-one cents,

how many pennies, nickels,

dimes, and quarters might I have?

Master 10-4: Calculator Test Items

Guidelines for Teaching Mental Computation

•Encourage students to do computations mentally.

•Learn which computations students prefer to

do mentally.

•Find out if students are applying written

algorithms mentally.

•Plan to include mental computation

systematically and regularly as an integral

part of your instruction.

• Keep practice sessions short, perhaps 10

minutes at a time.

• Develop children's confidence.

• Encourage inventiveness. There is no one

right way to do any mental computation.

• Make sure children are aware of the

difference between estimation and mental

computation.

Master 10-5: Guidelines for Teaching Mental Computation

Guidelines for Teaching Estimation

•Give your students problems that encourage

and reward estimation.

•Make sure students are not computing exact answers and then rounding to produce estimates.

•Ask students to tell how they made their

estimates.

•Fight the one-right-answer syndrome from the start.

•Encourage students to think of real-world

situations that involve making estimates.

Master 10-6: Guidelines for Teaching Estimation

Computational Estimation Strategies

Front-End Estimation

Adjusting

Compatible Numbers

Flexible Rounding

Clustering

Master 10-7: Computational Estimation Strategies

Mental Computation-Computation done internally without any external aid like paper and pencil or calculator. Often nonstandard algorithms are used for computing exact answers.

You drove 42 miles, stopped for lunch, then drove 34 miles. How many miles have you traveled? Explain how you solved the problem. 42 + 34

You earned 36 points on your first project. Then earned 28 points on your second project. How many points have you earned? Explain how you solved the problem.

36 + 28

You watched a video for 39 minutes. You watch a second video for 16 minutes. How many minutes did you watch in all? Explain how you solved the problem. 39 + 16

Computational Estimation-The process of producing

an answer that is sufficiently close to allow decisions

to be made.

You have $10 to buy detergent and a mop. Do you have enough? Explain how you solved the problem.

$ 3.98

+ 5.98

You have $5 to buy a soft drink, sandwich, and a slice of pie. Do you have enough? Explain how you solved the problem. $ . 68

2. 39

+2. 29

Master 10-8: Computing Mentally and Estimating

Three-Step Challenge

Use the , , =, and numeral keys on

your calculator to work your way from

2 to 144 in just three steps.

For example,

Step 1: 2 12 = 24

Step 2: 24  12 = 288

Step 3: 288  2 = 144

Solve this problem at least five other

ways. Record your solutions.

Choose your own beginning and

ending numbers for another three-step

challenge. Decide if you must use

special keys or all the operation keys.

Challenge a classmate.

How did you use estimation, mental

computation, and calculator

computation?

Master 10-9: Three-Step Challenge

A Student's View of Mental Computation

Interviews with students in several countries about their attitude toward mental computation produced surprising consistent responses. Here is a "typical" attitude of a middle grade student:

I learn to do written computation at school, and

spend more time at school doing written computation

than mental computation. I find mental computation

challenging, but interesting. I enjoy thinking about

numbers and trying to come up with different ways of

computing. It helps me to understand things better

when I think about numbers in my head. Sometimes I

need to write things down to check to see if what I have

been thinking is okay. I think it is important to be

good at both mental and written computation, but

mental computation will be used more as an adult and

so it is more important than written computation.

Although I learned to do some mental computation at

school I learned to do much of it by myself.

(McIntosh, Reys & Reys)

How would you respond to this student?

If you had an opportunity to talk with the student's teacher, what would you tell her?

Master 10-10: One View of Mental Computation